Definition
Let be a Lie algebra over a commutative ring R with universal enveloping algebra, and let M be a representation of (equivalently, a -module). Considering R as a trivial representation of, one defines the cohomology groups
(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor
Analogously, one can define Lie algebra homology as
(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor
Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.
Read more about this topic: Lie Algebra Cohomology
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)